Does anyone know of good references for nonstandard set theories and their applications to various branches of mathematics like category theory, algebra, geometry, etc.?

Edit: What I mean by "nonstandard set theory" is a formalization of the naive notion of sets that allows direct arguments about certain intuitive notions like infinitesimals and infinite integers without recourse to model theoretic constructions like ultrafilters and ultraproducts. Infinitesimal number is the usual application I keep seeing but I'm sure there must be other applications and that's the intent of my question. I'm not sure if this is precise enough.

Could you try and make precise what you mean by "nonstandard set theory"? It sounds like what you're thinking of is nonstandard analysis (which can be set up using perfectly standard set theory).
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Yemon ChoiNov 23 '09 at 8:27

The point is that set theory is pretty much esoteric nonsense in the first place, and we only use it because it's really the only way we can make sure everything is formal. We wouldn't want to use any set theory that invalidated previous results, so any other set theory is going to be roughly the same strength as some conservative extension of ZFC, or at the weakest, merely a conservative weakening, like ETCS or SEAR. There are differences in specifics (typed, untyped, etc.), but what's true in one of these set theories almost always has a translation to the others.
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Harry GindiNov 23 '09 at 8:49

@fpqc: I'm unclear what you mean by "any other set theory." ZFC + Projective Determinacy is not a conservative extension of ZFC (assuming you believe that ZFC is consistent), but why would you not count this?
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John GoodrickNov 24 '09 at 0:18

@fpqc: Ah, so you don't mean "conservative" in the standard technical sense. I was confused, because in logic the word is usually used as described in wikipedia page on "Conservative extension."
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John GoodrickNov 24 '09 at 1:14

4 Answers
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It seems that Lars Brünjes and Christian Serpé have a whole program for introducing non standard mathematics in algebraic geometry; they wish to play with non standard contructions, seen internally and externally (the interest of this game consists precisely to look at the non-classical logic (or internal) point of view and at the classical (or external) one at the same time). For instance, for an infinite prime number $P$, the ring $\mathbf{Z}/P\mathbf{Z}$ behaves internally like a finite field, while externally, it is a field of characteristic zero which contains an algebraic closure of $\mathbf{Q}$. Non-standard constructions can often be interpreted in a precise way as standard ones using ultraproducts and ultrafilters. Their purpose is to develop all the tools of classical algebraic geometry (homotopical and homological algebra, stacks, étale cohomology, algebraic K-theory, higher Chow groups...) in a non-standard way, in order to prove facts in the classical setting. Most of their papers can be found here (their papers contains more precise ideas on the possible interpretations and explanations). Brünjes and Serpé see non-standard mathematics as an enlargement of standard mathematics, and their work deals a lot in making this precise. However, they seem to have quite few concrete problems in mind. For instance, they have found sufficient conditions on cohomology classes to be algebraic in a very classical sense (see arXiv:0901.4853).

I think you are really asking about using toposes (or topoi (take your pick)) in place of one of your standard set theories. I am new to this stuff, but Mac Lane and Moerdijk's "Sheaves in Geometry and Logic" gives a very good introduction to thinking this way. As far as using infinitesmals go, I would really recommend Anders Kock's Synthetic Differential Geometry, available on his website for free: http://home.imf.au.dk/kock/.

The point is that toposes have their own internal logic, and you can reason about them as if they were a universe of sets in that logic. As the intuitionist try to tell us, the mathematical universe CAN look a lot nicer and more regular without the axiom of choice, and without excluded middle, so it is possible to construct toposes which have a lot less pathology than the ordinary universe of sets. These toposes can then support things like infinitesmals.

This is not what I would call "nonstandard", but you may be interested in surreal numbers. They were originally developed for combinatoric game theory but they allow somewhat unique circumstances in treating infinity and infinitesimals; like the paper below explains, using surreal numbers you can discuss "the square root of infinity" without it being complete nonsense.

A surreal number is a pair of sets of surreal numbers L (the "left set") and R (the "right set") such that no member of R is less than or equal to any member of L. Traditionally the numbers are written { L | R }.

Given surreal numbers x and y, $x \leq y$ if and only if y is less than or equal to no member of x’s left set, and no member of y’s right set is less than or equal to x.

We define the surreal number {|} to be 0.

These definitions (along with logical ones for addition, subtraction, etc.) spin out an entire number system, and it ends up that one version of $\epsilon$ is {0 | 1, 1/2, 1/4, 1/8, ...} and one version of $\omega$ is {$\mathbb{Z}$ | 0}.

While having infinity and infinitesimals as actual numbers in the system sounds like a good deal, it makes integration or differentation hard (I don't believe anyone has yet found a method).